Right now, the lowest-price time of year to take a flight to Taipei is March/2020. June/2020 is the most expensive time to fly to Taipei.
EVA Air, Qantas Airways and China Airlines fly non-stop from Australia to Taipei.
The cheapest airline that flies to Taipei from Australia is Scoot.
The best airline to fly from Australia to Taipei is Singapore Airlines. EVA Air is the best if you need non-stop flights. AirAsia X is the best budget airline.